Indifference Curves
Brennan T. Beal, PharmD
Impetus For The Post
When I first learned about indifference curves, I was shown a monstrosity that you all may be familiar. It looked a bit like the image below. If you were lucky, you were shown a computer generated image. The less fortunate had a professor furiously scribbling them onto a board.
A few things were immediately of concern: why are there multiple indifference curves if it only represesnts one consumer? Why are the curves moving? And… who is Natasha? So, while answering my own questions, I thought sharing the knowledge would be helpful. This post will hopefully provide a better description than maybe most of us have heard and by the end you will understand:
What indifference curves are and what they represent
How a budget constraint relates to these indifference curves
We will discuss how to optimize utility within these constraints (if you’re brave)
Finally, (because I am officially interested in healthcare economics) how this all relates to healthcare
For the scope of this post, I’ll asssume you have some fundamental understanding of utility theory.
Indifference Curves
An indifference curve represents a function which describes all the different combinations between two goods that yields the same amount of utility (i.e. to which a person would be indifferent). So, an indifference function defined by \(U(x,y) = 2x + y\) would mean that a person would be have an indifference to two “\(x\)’s” or one “\(y\)”. In more practical terms, let x represent quantity of wine while y represents a quantity of cheese. If the same utility function from above applies, then a consumer would be indifferent between two units of wine and one serving of cheese.
So, that is the thought behind indifference curves - combinations of quantities between two goods that yield the same utility. Notably, there could be any number of goods within a utility function but we will stick to two (\(x\) and \(y\)) for grpahical purposes. But we still have some things left wanting - why are there multiple on the same coordinate plane (remember the plots from above)?
Something that was not abundantly clear to me is that the left side of an equation represents a \(z\) axis. In theory, someone could have any amount of utility that they can afford - we will get to that in a bit. And intuitively, a consumer will always prefer to maximize their utility. With this in mind, multiple curves on a single plane are really just projections from a 3-dimensional graph.
The utility function actually looks like this (this equation is known as the Cobb-Douglas function and it was originally created to describe production): \[U(x,y) = Ax^{a}y^{1-a} \]
font_list <- list(
family = "Helvetica, sans-serif",
size = 10,
color = "#4b2e83"
)
x_axis <- list(
title = "Units of Wine",
titlefont = font_list
)
y_axis <- list(
title = "Units of Cheese",
titlefont = font_list
)
z_axis <- list(
title = "Utility",
titlefont = font_list
)
p <- plot_ly() %>%
add_surface(z = ~goods_matrix, showscale = FALSE,
contours = list(
z = list(
show=TRUE,
usecolormap=TRUE,
highlightcolor="#ff0000",
project=list(z=TRUE)
),
size = 90
)) %>%
layout(
showlegend = FALSE,
scene = list(
camera=list(
eye = list(x=1.87, y=0.88, z=1)
),
xaxis = x_axis,
yaxis = y_axis,
zaxis = z_axis
)
) %>%
config(displayModeBar = FALSE)
div(p, align = "center")
Now that you’ve seen the indifference curves, here are a few important concepts to cover here regarding assumptions and properties of the utility function and the implied indifference curves.
Properties and Assumptions of Indifference Curves
Indifference curves are downward sloping: this means that, from left to right, or as x increases, y must necessarily decrease. The property is derived from the fact that if a consumer is truly indifferent to some bundle of goods, an increase in one will mean a decrease in another. Further, this slope is known as the “Marginal Rate of Substitution” (MRS), which means that the slope at any given point represents the consumers willingness to substitute his current bundle for another. Mathematically: \[MRS = \delta y/\delta x\]
Diminishing Marginal Rates of Substitution: implies that as you gain more more of one good, you are willing to give up less of the other. Further, the negative slope of our indifference curve. As the combinations of \(x\) and \(y\) increase, utility increases. Hopefully now the intuition behind the indifference curves (really, indifference surfaces) are clear. Multiple curves represent the topography of a given utility function.
Higher curves are preferred to lower curves: intuitively, a consumer will seek to maximize their utility and will do so to the extent possible (usually within some given constraint). This also means that if a consumer has access to a certain amount of utility, the area under the curve is preferred less to points along the curve (because that area represents a lower utility, topographically).
Preferences are transitive: assumes that each point on a higher indifference curve is preferred to any point on a lower curve. Further, if all points on \(IC_3\) is preferred to points on \(IC_2\), and all points on \(IC_2\) are preferred to points on \(IC_1\), then all points on \(IC_3\) are preferred to \(IC_1\) - this along with the negatively slopping lines indicate that various indifference curves may never cross.
With this knowledge, we can now talk about the budget constraint and how it interacts with our indifference curves.
Budget Constraints
As I said earlier, utility theory suggests that one will seek to maximize their utility in all situations. So, naturally, their must be some constraint. And it isn’t hard to imagine that the constraint is often a given budget. In keeping with our example, we can note that a consumer has some indifference towards wine and cheese within the correct combinations (see the \(MRS\) that we spoke of earlier). Further, utility would be the highest with an infinite amount of both… but we know, in reality, this dcannot be the case (for most of us).
A budget, then, tells us how much we can spend so that we must maximize our utility within that constraint.
Let us use \(z\) again to denote our budget and assume that wine is half the amount of cheese. In this case, the price of one unit of wine, \(P_x\), is $0.25 and the price of one unit of cheese, \(P_y\), is $0.50 (we are living in a golden age). Your budget is then defined as \[I < (1/4)x + (1/2)y\]
This budget function could be anything and the coefficients represent prices. In this case, I can afford 2 units of cheese to one unit of wine and my budget constraint, \(I\), will tell me how much of each I can buy. Additionally, the ratio of prices, \(P_y/P_x\), represents the slope of the budget function.
In reality, the budget function, similar to a utility function, looks something like this (depending on the ratio of prices for two goods):
# --------- Creating Plot_ly
font_list <- list(
family = "Helvetica, sans-serif",
size = 10,
color = "#4b2e83"
)
x_axis <- list(
title = "Units of Wine",
titlefont = font_list
)
y_axis <- list(
title = "Units of Cheese",
titlefont = font_list
)
z_axis <- list(
title = "Budget",
titlefont = font_list
)
b_plot <- plot_ly() %>%
add_surface(z = ~budget_matrix, showscale = FALSE,
contours = list(
z = list(
show=TRUE,
usecolormap=TRUE,
highlightcolor="#ff0000",
project=list(z=TRUE)
),
size = 90
)) %>%
layout(
scene = list(
camera=list(
eye = list(x=1.87, y=0.88, z=1)
),
xaxis = x_axis,
yaxis = y_axis,
zaxis = z_axis
)
)
div(b_plot, align = "center")As your budget increases, \(z\), so does your ability to purchase larger quantities of each item. But a consumer is limited to the budget she has. And, similar to the consumers utility: we assume that a consumer maximizes their budget (so that \(budget = (1/4)x + (1/2)y\)) and that everything below the curve is considered suboptimal. As a persons budget increases, the line shifts right, and left as a budget decreases.
Now the question remains, how to we maximize utility within a given budget constraint if we know the functions that define both?
Maximizing the Utility Function
Marginal Rate of Substitution = Ratio of Prices
The first, and simplest method, is to find the point where the budget constraint, \(budget = (1/4)x + (1/2)y\), and the utility function, \(U(x,y) = Ax^{a}y^{1-a}\), are tangent. In other words, the point on both graphs where the slopes the slopes equal.
Fortunately, we have already learned that the slope of our utility function is equivalent to the \(MRS\), which is equivalent to: \[MRS = \frac{\partial u}{\partial x}/\frac{\partial u}{\partial y}\].
And we know that the slope of our utility function is just the ratio of the prices: \[P_x/P_y\]
Then, with a few pieces of information (\(P_x = 1/4\), \(P_y = 1/2\), a given budget (\(I = 50\)), and a given utility function \(U(x,y) = x^{.6}y^{.4}\)), we can derive the following using some simple partial derivatives and a system of equations:
\[\frac{1/4}{1/2} = \frac{\partial u}{\partial x}/\frac{\partial u}{\partial y}\]
For the right side of the equation: \[\frac{\partial u}{\partial x} = .6x^{-.4}y^.4\] \[\frac{\partial u}{\partial y} = x^{.6}.4y^{-.6}\] \[\frac{\partial u}{\partial x}/\frac{\partial u}{\partial y} = \frac{.6x^{-.4}y^.4}{x^{.6}.4y^{-.6}} = \frac{3y}{2x}\]
Now we can show: \[\frac{1/4}{1/2} = \frac{3y}{2x}\]
And with our systems of equations:
\[50 = (1/4)x + (1/2)y\] \[\frac{1/4}{1/2} = \frac{3y}{2x}\]
We find out that the optimal values to maximize utility within our budgete constraints are: \[ y = 40, x = 120\]
Lagrangian Multipliers
The more flexible, but slightly more mathematically intense option, is to use what is know as a Langrangian multiplier of the utility function with a given constraint (in this case, our budget constraint). Langrangian multipliers are, graphically, a way to find local minima and maxima within any given constraint so it works well in our context.
The formal equation reads: \[\mathcal{L}(x, y, \lambda) = U(x,y) - \lambda(xP_x + yP_y - I) \]
The equation requires the partial derivation of \(\mathcal{L}\) with respect to \(x\), \(y\), and \(\lambda\) and so we won’t go through that here. Find that in Appendix A. For now, know that we arrive at the same conclusion given the constraints described at the beginning of this section.
\[ y = 40, x = 120\]
See for yourself!
knitr::include_app("https://brennanbeal.shinyapps.io/indifference_curves/",
height = "800px")A work by Brennan T. Beal
brennanbeal@gmail.com
